I don't accept your authority on the argument of authority, guess that means you are wrong. — Jeremiah
I may have been off on appeal to authority, but that is not a reason to sweep aside 100 years of history, especially when we are talking about language, if people have been calling it a paradox for over 100 years, guess what it is a paradox. — Jeremiah
Also you forgot the link to the OED, which provided the definition of a paradox. Try reading it, as it turns out contradictions can be paradoxes. Is that the argument I made that you cut out? Was that what I was trying to say with the link? Because it is an authority. Appeal to authority, is not a reason to shrug off an valid authority. — Jeremiah
I didn't mistake anything. — MindForged
If people have been calling it a paradox for over 100 years, guess what, it is a paradox. — Jeremiah
Why is it that in the case of (a) you regard this as a basic mathematical truth; yet in the case of (b) you regard this as a philosophical conundrum perhaps susceptible to attack via paraconsistent logic? — fishfry
I assume (although you have not confirmed this) that you don't regard the infinitude of primes as being subject to modification or revision based on paraconsistent logic. Why is (b) different? — fishfry
Assume the contrary, derive a contradiction, learn a truth. — fishfry
You can't resolve a paradox but simply stating that it is not a paradox. A paradox by any other name is still a conundrum. — Jeremiah
I just don't have a lot of free time — Jeremiah
I actually already addressed this argument of yours. — Jeremiah
Proof by contradiction would lead us right back to Russell's Paradox.
It seems you have another contradiction on your hands. — Jeremiah
By the way, are you and/or MindForged making some kind of constructivist or intuitionist argument that rejects the law of the excluded middle and/or proof by contradiction? — fishfry
I think I've already articulated my position without recourse to intuitionism. — MindForged
Once you think it over (need not agree obviously) let me know what you think. — MindForged
I'll admit, I'm something of a logical pluralist so it's not like I'm advocating a wholesale abandonment of standard maths. Honestly, I actually wonder what mathematicians who think about this sort of thing believe (rare-ish to see it done in depth, most don't bother with the foundations of maths these days). Really, it seems like Gödel's Incompleteness Theorems in particular and the death of Logicism (using classical logic) seems to have killed foundationalism in the eyes of mathematicians and logicians, so I wonder if they're pluralists of a sort? — MindForged
Ok. Just wanted to make sure you accept law of excluded middle and proof by contradiction. — fishfry
Hamkins to see what the set theorists are up to. But nobody worries about Russell's paradox because there's nothing to worry about. It just shows that we can't use unrestricted set comprehension. And I still don't know why you think people should be concerned about a run of the mill proof by contradiction. Sure it ruined Frege's day, but it revealed a mathematical truth about the nature of sets. — fishfry
What do you think? — MindForged
I'm not talking about Russell's Paradox in that bit, I'm talking about the general outlook regarding mathematics post-Incompleteness Theorems. ZFC's development was intentionally practical: we need to get on with the business of doing sensible maths but classical logic cannot function sensibly with an inconsistent set theory. — MindForged
Once it became clear that there was no strict necessity in picking one formalism over another (i.e. no privileged set of indubitable axioms), it seems like mathematicians and logicians became a bit more cavalier about the whole thing. Rightly so, in my view, the interest shifted to the virtues of particular formal systems applied in specific domains, particularly when such systems are fruitful. — MindForged
Like from the Incompleteness Theorems, we know you can (for systems expressive enough to articulate arithmetic truths) either have an inconsistent but complete mathematics (Paraconsistent mathematics) or you can have a consistent but incomplete maths (Classical math, Intuituonistic math, etc.). Classical logic is so preferred because of its wide usability, but there are known issues and domains where it's questionable (quantum mechanics, representing human reasoning, databases, some evidence paraconsistent logic operations are faster to compute, etc). — MindForged
So I wonder if this modern openness to more or less any non-trivial logic/math indicates some kind of pluralism. What do you think? — MindForged
learn a truth — fishfry
I don't see why. Classical logic goes back to Aristotle. And even math doesn't need set theory. There wasn't any set theory till Cantor and there was plenty of great math getting done before that. Archimedes, Eudoxus, the medieval guys Cardano and so forth, Newton, Gauss, Euler, Cauchy, and all the rest. None of them ever heard of set theory and did fine without it. — fishfry
Incompleteness is literally a classical result now. Everyone's moved past it. So we can't use the traditional axiomatic method to determine what's true. If anything, that's perfectly sensible. We have to find other paths to truth. That's exciting, not worrisome I think. — fishfry
Now I certainly didn't say it resulted in nihilism, and I don't deny good math is being done. As I said, I don't reject standard math.I really don't believe that incompleteness is any kind of nihilistic disaster. Interesting math is being done every day. — fishfry
On the other hand, perhaps it's related to postmodernism and the reaction against reason. — fishfry
The axiom schema of specification blocks Russell. Would I be right in thinking that one reason to be cool with that approach (the truth learned) is that we don't need unrestricted quantification? — Srap Tasmaner
No, Aristotle created Syllogistic. Classical logic was invented in the 1870s by Frege. These are not the same system, — MindForged
Would I be right in thinking that one reason to be cool with that approach (the truth learned) is that we don't need unrestricted quantification? — Srap Tasmaner
it has been demonstrated that naive set theory + a paraconsistent logic lets you prove the Continuum Hypothesis is false. — MindForged
"It should also be noted that Brady’s construction of naive set theory opens the door to a revival of Frege-Russell logicism, which was widely held, even by Frege himself, to have been badly damaged by the Russell Paradox. If the Russell Contradiction does not spread, then there is no obvious reason why one should not take the view that naive set theory provides an adequate foundation for mathematics, and that naive set theory is deducible from logic via the naive comprehension schema.
(...)
Even more radically, Weber, in related papers (2010), (2012), has taken the inconsistency to be a positive virtue, since it enables us to settle several questions that were left open by Cantor, namely, that the well-ordering theorem and the axiom of choice are provable, and that the Continuum Hypothesis is false (2012, 284)."
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